Page 213 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

August 31, 2006
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JWBK119-13
198 A Graphical Approach to Obtaining Confidence Limits of C pk
limits are termed ‘nonconforming’ (NC). An indirect measure of potential capability
to meet the settings LSL, X, and USL is the process capability index
USL − LSL
C p = . (13.1)
6σ
Here σ denotes the standard deviation of Xand is estimated by

n
1
¯
2
s = (X j − X) , (13.2)
n − 1
j=1
where n is the sample size and
n
1
¯
X = X j . (13.3)
n
j=1
Denoting the midpoint of the specification range by m = (USL + LSL)/2, the shift
index k is given by
|m − μ|
k = . (13.4)
(USL − LSL) /2
The proportion of NC product can be estimated in term of C p and k by
ˆ ˆ
ˆ ˆ
ˆ p = [−3(1 + k)C p ] + [−3(1 − k)C p ]. (13.5)
A contour plot of equation (13.5) on two different scales (Figures 13.1 and 13.2)
reveals some interesting and useful properties. It can be seen from the figures that
while (C p , k) uniquely determine p, there also exists a unique value of k for each
(C p , p). Intuitively, we know that for each constant k, p increases as C p decreases, and
for each constant C p , p increases as kincreases. Figure 13.1 also reveals that for each

p = 0.25 0.20 0.15 0.10 0.08 0.06 0.04
0.7 0.02
0.01
0.6 0.005
1.0e−03
0.5
1.0e−04
0.4
k
1.0e−05
0.3
1.0e−06
0.2 1.0e−07
p = 1.0e−08
0.1

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
C P
Figure 13.1 Behavior of C p , k and p on linear scale.